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17
TUNNELING IN MULTIDIMENSIONAL WELLS
T.F. Pankratova ITMO University, St. Petersburg, Russia tanpankrat@yandex.ru
PACS 32.30-r; 03.65-w; 73.21.Fg; 78.67.De; 31.15-xr; 05.45.xt DOI 10.17586/2220-8054-2015-6-1-113-121
A full asymptotic series for low eigenvalues and eigenfunctions of a stationary Schrodinger operator with a
nondegenerate well was constructed in [29]. This allowed us to describe the tunneling effect for a potential with
two or more identical wells with sufficient accuracy. The procedure is described in the following discussion. Some
formulae are obtained and corresponding problems are discussed.
Keywords: Shrodinger operator, potential, tunneling, eigenvalues and eigenfunctions.
Received: 21 October 2014
Revised: 20 November 2014
1. Introduction
Consider the Schrodinger equation:
h2
— — Au + Vu = Eu, (1.1)
where A = t=i d2/dx 2 is the Laplace operator, V is a real valued function defined on Rd having nondegenerate minima (wells) with some mode of symmetry.
If V has a finite number of identical wells which differ only by space translations and V(x) > C beyond the region of the wells where C exceeds the value of V at minimum, the lower part of the spectrum of the corresponding Schrodinger operator is organized in the following way. There is a set of finite groups of eigenvalues (each connected with some quantum vector n G Nd), the distance between the groups being of the order h, and the distance between eigenvalues in each group, the splitting, being exponentially small with respect to h.
It is possible to find explicit formulae for the widths of these splittings using semiclas-sical asymptotics for each well. The problem was considered in different ways by different authors and almost completely solved in the one dimensional case [1-11]. The case d > 1 seems much more complicated. There are many results obtained in this area (see [11-20] and the list is far from complete). Still, the picture is not so complete as when d =1. The semi-classical asymptotics of the discrete spectrum and strict estimates of the splittings are described in [11-13] and other works of these authors (using the theory of pseudo differential operators). The semiclassical expansion for the eigenfunctions and the rigorous asymptotics for the splitting widths in the lowest levels (n = 0) were obtained in [18-20] (with the use of a Maslov's canonical operator). Still, there are no effective (as when d =1) splitting asymptotic formulae in terms of the potential for a set of arbitrary levels (|n| = 1, 2, 3,...). The possibility of solving this problem in that case was discussed on the Diffraction Day Conference this spring in the talk of A. Anikin and M. Rouleux. These results have not been published yet.
My approach to this problem is different. In order to write down strict asymptotic formulae for splittings in the d-dimensional case, one has to develop methods of [9]. To do
so, it is necessary to find a sufficiently accurate semiclassical approximation to eigenstates for a single well in some vicinity of a minimum, independent of h. Such an approximation was constructed in [29]. The formal series on powers of h was obtained. Coefficients in all terms were found in some domain independent of h. Terms for eigenfunctions are analytic for analytic potential. If we truncate the series at the N-th term, the remaining sums satisfy the equation (1.1) with an error on the order of hN+1 exp(-S/h) , where S is a nonnegative function defined in [29]. They give us so called quasi-modes [21]. The possibility to take N as large as we like and exponential decreasing of all terms beyond some vicinity of a minimum allows one, with the help of quasi-modes, to find real eigenfunctions and eigenvalues approximately, with exponentially small errors, smaller than the widths of the splittings. (The program was realized in [9] for d =1.) The constructed series allow us to investigate the set of zeros for the eigenfunctions. The latter is interesting by itself and may be essentially used while finding the splitting asymptotics for |n| > 1.
2. Asymptotic expansions for the eigenstates in one well
We look for eigenfunctions un and eigenvalues En of (1.1) where V is a real valued function defined on Rd having a nondegenerate minimum at the origin in the form of the following series:
E„
un = exp
X
/ Enjh j j=1
^ ^ unj h j j=0
where Enj e R, n = (n1,n2,..., nd) e Nd is a quantum vector, S = S (x), x = (x1, x2,. Rd, unj — u nj(x), j = 0,1, 2,..., are functions independent of h. One can find un0 in the following form:
uno = iJnePn (x),
where:
^ = ^ (x) = M (x) (x) j... (x)) j r = n€
i= 1
(2.1)
(2.2)
,xd) G
(2.3)
the functions (x), i = 1,2,... ,d, and S(x) satisfy the following equations:
1
d
S (x) = s0 (x) = - £ €2
i=1
SJ =-E(1 - 2^ij )€i2j j = 1j...jdj
i= 1
8.
ij
1j if i = jj 0, ifi = j.
(VSj
2
2V,
j = 0,1j...jdj
(2.4)
(2.5)
(2.6)
<V€i, V€j > = 8ij (V€j )2. (2.7)
Symbols V and <■, ■> denote a gradient and a scalar product in Rd respectively. We put the series (2.1) and (2.2) into the Schrodinger equation (1.1) and equate coefficients of each power of h to zero. The equation for power 0 is satisfied automatically because
of (2.6). The requirement for the coefficient of first degree in h to be equal to zero, gives us the following equation for the function Pn and the number En1:
AS° d
<VS°, VP„> =Eni--2--S ni(Wi)2. (2.8)
i=1
The analogous requirement for the coefficient of h2 gives the equation for un1 and En1:
<VS°, Vuni) + (Af--En^ Uni= j
APra+(VPra)2 ,
-^--rEn2
ePn, (2.9)
where
d / ^ d d
Fi = £ ni(n2- X) c*-1 )2 n j + E n + E ^r1 (V^i, VPn) n j i=1 j=i i=1 i=1 j=i
So on, for each j > 2 we obtain the equation:
<VS°, Vunj) + ^A2S- - Eni^ unj
Aun,j-i 1
+ V En;i+iun)j-i + ^nePn(x)En,j+i. (2.10)
2
1=1
In [29], a procedure of constructing solutions for these equations was described and the following theorems were proven.
3. The phase theorem for the analytic potential
Let V be analytic with the following Taylor series:
d d V (x) = 1 E Ux2 + E Vkxk, k = (ki, k2,..., kd) G Nd, |k| = E k , (3.1)
i=1 |k|>3 i=1
convergent in a polydisk |x^ < r, i = 1, 2,..., d with the numbers u > 0, i =1, 2,..., d. We search for solutions for equation (2.6) in the form of a power series:
1 d
(x) = 2 E U (1 — )x2 + E jxk, j = 0,1,..., d, (3.2)
i=1 | k|> 3
and comparing coefficients of xk , we find the following recurrent formulae for (Sj )k:
)»= j' (33)
where u = (u^ u2,..., ud); /0 is a unitary matrix of order d; /j, j = 1,..., d, is a diagonal matrix of the order d with —1 standing at the j-th place of the diagonal and 1 at the others, Vk = vk for |k| = 3 and Vk = vk + terms, depending on (Sj)z, |/| < |k|, for |k| > 4.
It is easy to see that for the positive numbers u», i = 1, 2,..., d, the denominators in expressions (3.3) for j = 1, 2,..., d, can be equal to zero. So, even to formally construct these series, we have to impose some additional conditions on the potential V.
Simultaneously, we construct a change of variables:
: Vj = (Vj1,... , Vjn) ^ (xi = (Vj) ,... , xd = $d(yj)) , (3.4)
which transforms the vector field (VSj, V-) into the normal form:
d d L° = J] ^i (1 - 2Sij) Vijd—. (3.5)
i=i Vij We search the functions (y), i =1, 2,..., d, in the following form:
(Vj) = Vij + )fcvk,. (3.6)
|fc|>2
In order to find the coefficients ($j)fc, we replace xi, i =1, 2,..., d, in (VSj, V-) by $i (Vj) of the form (3.6) and equate the obtained series (in variables y) to L°. Hence, we find the following expressions for the coefficients:
(<tj )k = (k-TSTj ,j =0 1--d- |k|>2 (37)
where:
Sj = (ki + 1) (Sj)fc+ort. + terms, depending on ($j)m , l = 1, 2,..., d, |m| < |k| ,
orti is an element of a standard basis {ort^ having all components equal to 0 except of the i-th one, which is equal to 1.
We see here that some denominators are equal to zero for some values of w. We have to exclude these values.
Let us make the following definitions:
(1) we say, that the positive numbers w^ w2,..., wd are nonresonant- if they are linearly independent over integers;
(2) positive numbers w^ w2,..., wd are said to be Diophantine if there exist positive numbers a and C such that for any k e Zd, k =0,
|(k,w)|>^; (3.8)
(3) we denote the set of vectors w = (w^ w2,..., wd) with positive components by Q, the set of w with nonresonant components by Qnr, the set of w with Diophantine components by .
Theorem A. Let the potential V be analytic, represented by a series of the form (3.1) convergent in the vicinity of the origin.
(1) If e Qnr, then there exists a pair: a unique positive analytic function S which can be represented by convergent series of the form (3.2) for j = 0 in some vicinity of the origin and satisfies the equation (2.6); and a unique analytic diffeomorphism which transforms the vector field (VSj, V-) to the normal form L° given by (3.5).
(2) If e , then for each j e {1, 2,..., d} there exists a pair: a unique analytic function Sj which can be represented by convergent series of the form (3.2) in some vicinity of the origin and satisfies the equation (2.6); and a unique analytic, diffeomorphism $j which transforms the vector field (VSj, V ) to the normal form L° given by (3.5).
The proof of this theorem is published in [29]. Remark 3.1. Normal forms of the vector fields (i.e. of Hamiltonian systems of differential equations) are described in literature on classical mechanics e.g. [22-25]. A typical situation there is that given a vector field, one has to find the simplest form for it in suitable variables. Here, we have no given vector fields. We are looking for vector fields which are solutions for the nonlinear Eiconal equation (2.6). The normal forms (3.5) are used as an auxiliary tool.
Remark 3.2. In case (1), the nonresonance condition is necessary to construct (not S0). There are no small denominators in (3.3) for j = 0. The existence of analytic S° was established in [26] in a more general situation.
Remark 3.3. One can give the following geometrical interpretation for the results of the Theorem A. The functions Sj are the generating functions for Lagrangian manifolds, which are invariant with respect to the classical dynamical system with the potential — V(x). The potential —V(x) has a 'hunch' at the origin (instead of a 'well' of V(x)). So our quantum mechanical problem 'at the bottom of a well' is equivalent to a classical problem 'near the top of a hill'. The origin is a point of singularity in this problem, a point of infinite time in classical dynamics, a point of vanishing energy of the Lagrangian manifolds. The theorem gives the existence of the generating functions SJ for the invariant Lagrangian manifolds in a small vicinity of that point.
The geometrical aspects of the problem were considered in [27]. In addition to the proof of Theorem A, the following lemma was proven.
We denote by Br a Banach space of analytic functions in B (r) = y G Cd : |y»| < r, i = 1, 2,... ,d with the norm ||f || = supyeB(r) |f (y)| , by Br)M,n,° the subspace of Br which is the set of functions having the Taylor series, which contains only the terms with power |k| > M > 0 and coefficient at power n equal to zero, by u° = minie{1)...)d}ui. Lemma 1. Given r' < r, there exists a bounded operator {L^} : Br)Mra 0 ^ Br/)Mra 0 which solves the equation L^ = f |B(V), u G Br/)M ra 0, f G Br>M,n,°, in the following cases:
(1) for any u G Q, j = 0, n = (0,..., 0);
(2) for uG Qnr, j = 0, n arbitrary;
(3) for uG , j = 1,..., d, n arbitrary,
and there exists a positive constant c1 = c1 (M, d, u, r) such that in both cases (1), (2):
K}-
-1
< -
( r — r ' ) d- 1
in case (3) there exists a positive constant c2 = c2 (a, M, d,u, r) such that:
< —C1
K}
4. Constructing the series (2.1), (2.2)
1
< C2
(r — r')
a+d '
In order to construct the whole series (2.1) and (2.2), we have to find at first (after solving (2.6)) all the functions Vj (x) which satisfy the following equations:
V2 (x) = S° (x) — Sj (x) , j = 1, 2, ... , d . (4.1)
Lemma 2. Let Sj (x), j = 0, ..., 1, d, be taken from Theorem A.
Then, the right hand sides in the formulae (4.1) are the full squares, i.e. there exist d unique analytic functions Vj, j = 1, 2...,d, which satisfy the equations (4.1) and have the following convergent series:
^ = Vj + J] (^xk, j = 1, 2, ... , d, (4.2)
|fc|>2
in iome vicinity of the origin.
After changing the variables of (3.4), the equation (2.8) satisfies the conditions of Lemma 1, case (1), M = 1, if we choose En1 in the following way:
Eni = ^ (n + ^ (4.3)
According to Lemma 1, there exists an analytic solution, which after returning back to coordinates x, gives us in some polydisk an analytic solution for (2.8) which vanishes at the origin.
Each of the equations (2.9) and (2.11) has the following form:
/ A S0 \
<VS°, Vuni) + i — - En A Uni= F. (4.4)
We search for the solution of (5.7) in the form of the product:
u = UePo, (4.5)
where P° is a solution of equation (2.8) for n = 0. This means that:
VS°, Vepo) + (^S0 - ePo = 0. (4.6)
After putting (4.5) into (4.4) we obtain the following equation for the unknown function U :
L°U -<n,w) U7 = Fe-Po, (4.7)
where L° = L° is the normal form of the operator <VS°, V-) in coordinates y, 'tilde' means the change of variables: F (x) = F(y). Now the left hand side operator is that of Lemma 1, case (2).
The condition of solvability for equation (4.7) is the following:
Fe-Po) = 0, (4.8)
n
(F)n is noting the Taylor coefficient at yn of the function F.
Hence, we obtain the following expressions for all the terms of the series (2.1), i.e.:
EnO — — n2 2
— u 2
2
APra + (VPr
0
- u-FiePn-Po) , (4.9)
n
AUj-i ^
/ y En,l+1un,j-1
2 l=1
e-Po ) , j > 2; (4.10)
and find all the functions wnj, j — 1, 2,... in form (4.5).
n
5. Main theorem and concluding remarks
Results from paper [29] are summarized in the following theorem.
Theorem B. Let the potential V in Schrodinger equation (1.1) be analytic, represented in a vicinity of the origin by Taylor series (3.1) with positive Diophantine numbers ...,
Then for any G Nd, 0 < |n| < n*, N G N, one can construct the following pair: a number:
N
^ ^ h )
and an analytic function:
/ s0 \ N-1 h
expj —hj
which satisfies the Schrodinger equation (1.1) up to terms of the order hN+1exp { —S0/h} in some vicinity of the origin independent of h. Here: S0(x) is the positive analytic solution of (2.6) with Taylor series (3.2) (see Theorem A), analytic functions (x) (j = 0,1,...,N — 1) and numbers (x) (j = 1, 2,...,N) have the form given by formulae (2.3), (4.2), (4.5) and (4.3), (4.9), (4.10).
Remark 5.1. One can lengthen the functions SJ analytically onto a larger domain by the formulae Sj = j ^d=1 pjdxj, j = 1, 2,..., d, where for each j the integral is taken along the trajectory of the corresponding Hamiltonian system. Hence, one can lengthen the functions Vj (x), j = 1, 2,..., d, and uraj, j = 0,1, 2,... in a similar way. Thus, one can construct sufficient quasi-modes in a rather large domain containing the point of a minimum. Then, in the problem with many identical wells, situated so that the distances between the points of the minima are finite, one can do the following. Construct quasi-modes for each well in such a domain, that the two neighboring domains intersect. Then multiply those quasi-modes on the cutting functions equal to zero beyond the mentioned domains. The approximation for the eigenfunctions of the problem can be taken as a linear combination of these cut-off quasi-modes. It is then possible to write the rigorous splitting formulae following the ideology of [9] for an arbitrary n G N in the form as it was obtained in [18-20] for n = 0.
Thus, one can find:
AE„ = a„e"bn/h (1 + O(h)),
where, in the case d =1, = JJ2 \J2V(x)dx,
in the case d > 2, = Jmim2V2V(x)d/, the last integral is taken along the extreme line of the functional in the right hand side of the last formula.
It is important to note that to find the pre-exponential coefficient in the splitting formula for |n| > 0, one has to be sure that on the trajectory of the corresponding Hamiltonian system the corresponding eigenfunction is not equal to zero. Hence, one has to investigate the zero-sets of the eigenfunctions. Some examples in a one-dimensional case were considered by my student N. Homchenko and published in [30].
Remark 5.2. In order to find the zero-sets of the eigenfunctions, one can also use expansions of the form (2.2). It is more convenient, however, to construct for this purpose an ansatz with Hermite polynomials, namely:
(5.1)
where S° and ^ i e (1,..., d), are the above-described Unctions, Hn. (t) = (—1)niet2 are Hermite polynomials which satisfy the following differential equation:
Hi (t) - 2tn; (t) + 2niHni (t) = 0.
(5.2)
If we put series (5.1) and (2.1) into the Schrodinger equation (1.1) and equate coefficients at each power of h to zero (taking into account (5.2)), we will obtain problems for Gj quite similar to those described in Section 4. Solving them, we will construct all the functions Gj. In zero approximation the eigenfunction un has the form of an exponent multiplied by a product of Hermite polynomials. Hence, in the zero approximation, we find a set of zeros of the function un as a net of intersecting surfaces Ej : ^ (x) = tj, i = 1, 2,..., d, tj e R^ Rj is a set of roots of Hn. (t). The first term of (5.1) depends on third and fourth derivatives of the potential V at the origin. It does not vanish if they are not equal to zero. In this case, one can already find in the first approximation, that Ej do not intersect. They have quasi-intersections. A more detailed description of this ansatz and some examples were published in [31] and [32].
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